I don't know if this has ever been suggested but I thought it would be neat. Here's my idea:

Chocobos scale to your current equipped class/job. They still have stances, however they now learn and use abilities automatically based on level. Take the current customization completely out to make room for...

"Chocobo Raising." Basically, you raise your Chocobo for side activities such as Racing and Treasure Dig Hunts with other players. It would be neat to have an area-wide event where everyone brings out their Chocobos and you dig with different abilities/attributes. It could be timed, or it would end when a specific item has been found. Racing is pretty straight-forward.

Rewards:
Bardings
Chocobo Stables/Accessories
Food and Breeding/Care Items
Vanity Gear

This is probably just a fantasy as it would take a massive amount of work and they already have something coming. But I just thought I'd share.